Arithmetic Homogeneous Spaces

The remarkable phenomenon of Superrigidity, discovered by Margulis in the context of linear representations of lattices in higher rank semi-simple groups, has motivated and inspired a lot of research on other "higher rank" groups and representations into target groups other than linear ones. In this joint work with Uri Bader and Ali Shaker, we propose a new approach to superrigidity, based on a notion of a "Weyl group" associated to a "boundary" of G (it becomes the Weyl group when G is semi-simple). We use this approach to prove various superrigidity results for representations into Homeo(circle), including an easy proof of Ghys' result, lattices in products of general lc groups, $\tilde A_2$ groups, cocycle versions of all the above, and commensurator superrigidity.